CAIE M2 (Mechanics 2) 2010 November

A horizontal circular disc rotates with constant angular speed \(9 \text{ rad s}^{-1}\) about its centre \(O\). A particle of mass \(0.05 \text{ kg}\) is placed on the disc at a distance \(0.4 \text{ m}\) from \(O\). The particle moves with the disc and no sliding takes place. Calculate the magnitude of the resultant force exerted on the particle by the disc. [3]

\includegraphics{figure_2} A bow consists of a uniform curved portion \(AB\) of mass \(1.4 \text{ kg}\), and a uniform taut string of mass \(m \text{ kg}\) which joins \(A\) and \(B\). The curved portion \(AB\) is an arc of a circle centre \(O\) and radius \(0.8 \text{ m}\). Angle \(AOB\) is \(\frac{2}{3}\pi\) radians (see diagram). The centre of mass of the bow (including the string) is \(0.65 \text{ m}\) from \(O\). Calculate \(m\). [6]

\includegraphics{figure_3} One end of a light inextensible string of length \(0.2 \text{ m}\) is attached to a fixed point \(A\) which is above a smooth horizontal surface. A particle \(P\) of mass \(0.6 \text{ kg}\) is attached to the other end of the string. \(P\) moves in a circle on the surface with constant speed \(v \text{ m s}^{-1}\), with the string taut and making an angle of \(30°\) to the horizontal (see diagram).

1. Given that \(v = 1.5\), calculate the magnitude of the force that the surface exerts on \(P\). [4]
2. Given instead that \(P\) moves with its greatest possible speed while remaining in contact with the surface, find \(v\). [3]

\includegraphics{figure_4} A uniform beam \(AB\) has length \(2 \text{ m}\) and weight \(70 \text{ N}\). The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached to the wall at a point \(1.7 \text{ m}\) vertically above the hinge. The other end of the rope is attached to the beam at a point \(0.8 \text{ m}\) from \(A\). The rope is at right angles to \(AB\). The beam carries a load of weight \(220 \text{ N}\) at \(B\) (see diagram).

1. Find the tension in the rope. [3]
2. Find the direction of the force exerted on the beam at \(A\). [4]

A particle \(P\) of mass \(0.28 \text{ kg}\) is attached to the mid-point of a light elastic string of natural length \(4 \text{ m}\). The ends of the string are attached to fixed points \(A\) and \(B\) which are at the same horizontal level and \(4.8 \text{ m}\) apart. \(P\) is released from rest at the mid-point of \(AB\). In the subsequent motion, the acceleration of \(P\) is zero when \(P\) is at a distance \(0.7 \text{ m}\) below \(AB\).

1. Show that the modulus of elasticity of the string is \(20 \text{ N}\). [4]
2. Calculate the maximum speed of \(P\). [3]

A cyclist and his bicycle have a total mass of \(81 \text{ kg}\). The cyclist starts from rest and rides in a straight line. The cyclist exerts a constant force of \(135 \text{ N}\) and the motion is opposed by a resistance of magnitude \(9v \text{ N}\), where \(v \text{ m s}^{-1}\) is the cyclist's speed at time \(t \text{ s}\) after starting.

1. Show that \(\frac{9}{15-v} \frac{dv}{dt} = 1\). [2]
2. Solve this differential equation to show that \(v = 15(1-e^{-\frac{t}{9}})\). [4]
3. Find the distance travelled by the cyclist in the first \(9 \text{ s}\) of the motion. [4]

\includegraphics{figure_7} A particle \(P\) is projected from a point \(O\) with initial speed \(10 \text{ m s}^{-1}\) at an angle of \(45°\) above the horizontal. \(P\) subsequently passes through the point \(A\) which is at an angle of elevation of \(30°\) from \(O\) (see diagram). At time \(t \text{ s}\) after projection the horizontal and vertically upward displacements of \(P\) from \(O\) are \(x \text{ m}\) and \(y \text{ m}\) respectively.

1. Write down expressions for \(x\) and \(y\) in terms of \(t\), and hence obtain the equation of the trajectory of \(P\). [3]
2. Calculate the value of \(x\) when \(P\) is at \(A\). [3]
3. Find the angle the trajectory makes with the horizontal when \(P\) is at \(A\). [4]